2012 - Spring

General Homework

page 21: 1.1 - 1.5, 1.9, 1.11

page 76: 3.5,3.7 - 3.9

page: 105: 4.1 - 4.7, 4.10, 4.11, 4.17

page 139: 5.1 - 5.3, 5.5, 5.7, 5.8

page 171: 6.1 - 6.7, 6.10, 6.11, 6.17

page 213: 8.2, 8.5, 8.18

page 271: 9.1 - 9.3, 9.5 - 9.11

page 293: 10.2 - 10.5

page 312: 11.1, 11.2, 11.6, 11.8

page 484: 17.4, 17.5, 17.7, 17.8, 17.17

For interpolating polynomial, make up your own data and compute the polynomials(both Newton's and Lagrange). Also, consider the following problems:

page 573: 20.19, 20.20, 20.27, 20.35

page 627: 21.1a,c, 21.2a,c, 21.3a,c, 21.4a,c, 21.5a,c, 21.11a-d

page 750: 1 - 6 but not the analytic solution nor the methods we did not discuss in class. Also, it's enough to apply these methods once or twice on an interval. (I don't want to be responsible for any of you going crazy by applying them 47 times!)

Assignment 1

Due: Thursday, January 26, 2012 at class time

For these problems, calculate all results to 6 significant figures.

Problem 1: xi+1 = (xi + A/xi)/2 is an iterative formulation that will calculate the square root of a number A. They taught it to kids in the 4th grade back in the '60s before calculators were invented. We .... I mean ... they called it 'divide and average'. So, take A to be 225, and let x0 be 100; this is your initial value. Use the formula to calculate x1, x2, and x3. For each, compute Et and et. Is it getting better? Give the results for the 10th iteration.

Problem 3.6 on page 76. Use only 6 terms, not 20. Compute the errors only for the final sum (6 terms for each method). Is there a difference? If so, why?

Problem 4.4 on page 105.

Problem 5.5 on page 139.

Assignment 2

Due: Tuesday, February 28, 2012 at class time

For these problems, calculate all results to 6 significant figures unless otherwise specified.

Use the coefficient matrix, A, from the system in problem #10.6 on page 293 to calculate a LU decomposition. Round your entries to 3 significant figures. Now use this LU decomp to compute the inverse of A, A-1. (You may show the computations for the first column only.) Multiply A by A-1. How close was the answer to I, the identity matrix? Did the rounding effect the outcome?

Solve the same system stated in 310.6 on page 293 using Gauss-Siedel iteration (reformed Jacobian). Show computations for one iteration. Show the resulting vector of the 5th iteration. Can you guess the solution at this point?

Problem 17.6 on page 485.

Proble 17.12 on pabe 485.

Assignment 3

Due: Tuesday, March 13, 2012 at class time

Problem 17.9 on page 485. Disregard the instructions to plot.

Use the data from problem #18.5 on page 522 to construct the Lagrange form of the interpolating polynomial for that data. Simplify only the coefficients. The evaluate at x = 3 for an estimate of the function that generated the data.

Problem 18.6 on page 522.

Find the natural spline that passes through the points {(xk , f(xk))} 3k=0, on the graph of f(x) = x + 3/x, using the nodes x0 = 1, x1 = 2, x2 = 3, x3 = 4. Check the accuracy of your interpolant at the value x = 3/2. Check it again at the value x = 7/2.

Assignment 4

Due: Tuesday, April 17, 2012 at class time

In problems 1 and 2 below, refer to I given by

Compute an approximation to I using a single application of the Trapezoidal rule (h = 3), Simpson's rule (h = 3/2), Simpson's 3/8 rule (h = 1). Which is the best result? Choose an appropriate adjective to describe your first approximation and an appropriate adjective to describe your third approximation.

Now compute an approximation to I using the composite Trapezoidal rule by doubling the number of subintervals two times (final h = 3/4). Then use Romberg integration to produce a really good approximation to I using these approximations. Compare this value to your best in #1.

Now, for the next two questions, let O.D.E. refer to y' = y t-2, y(1) = 2

For O.D.E., using h = 0.5, compute an approximation to y(1.5) using Euler's method, then Heun's method, then the Midpoint method.

Now use RK-4 with h = 0.5 to generate an approximation to the solution to O.D.E. at t = 1.5 and t = 2.0. Now calculate the percent relative error to the exact solution given by y = 2exp((t-1)/t) at t = 1. Was it good???! (ans: yes)